Interactive physics simulator
Wave
Explore how disturbances transport energy through physical media. Switch between transverse and longitudinal modes to observe particle dynamics, verify the wave speed equation v = f λ, and study pulse reflections at fixed, free, and infinite boundaries.
Wave Physics Laboratory
Interact with the string or piston, adjust parameters, and observe live kinematics.
Live Wave Telemetry
- Frequency (f)
- 0.00 Hz
- Wavelength (λ)
- 0.00 m
- Wave Speed (v)
- 0.00 m/s
- Amplitude (A)
- 0.00 cm
- Medium Tension
- 0.00 N
Introduction to Mechanical Waves
A mechanical wave is an oscillation of matter that transfers energy and momentum through a physical medium (solid, liquid, or gas). Unlike particles or projectiles, a wave does not transport bulk matter. Instead, individual atoms or molecules in the medium oscillate locally about their fixed equilibrium positions, passing the disturbance on to adjacent particles.
The elastic restoring forces inherent to the medium are responsible for bringing displaced particles back to equilibrium, driving the wave forward.
The Wave Equation
The speed v of a periodic wave is directly related to its frequency f and spatial cycle length, or wavelength λ:
Where:
- Wave Speed (v): The speed at which the wave crests or compressions travel through the medium (measured in m/s). It depends solely on properties of the medium.
- Frequency (f): The number of oscillations completed by a point in the medium per second (measured in Hz). It is determined by the source.
- Wavelength (λ): The distance between two corresponding adjacent points on the wave that are in phase, such as two consecutive crests or compressions (measured in meters).
Transverse vs. Longitudinal Waves
Mechanical waves are classified based on the direction of particle oscillation relative to the direction of wave travel:
- Transverse Waves: Particles of the medium oscillate perpendicular to the direction in which the wave propagates. Examples include waves on a string and seismic S-waves. The maximum upward displacement is called a crest, and the maximum downward displacement is a trough.
- Longitudinal Waves: Particles of the medium oscillate parallel to the direction of wave propagation. Examples include sound waves and seismic P-waves. Instead of crests and troughs, these waves consist of alternating regions of high pressure (compressions) and low pressure (rarefactions).
Wave Speed in a Medium
A common misconception is that pushing a wave source harder or faster will make the wave travel faster. In physics, the speed of a mechanical wave is determined purely by the physical properties of the medium:
- On a string: The speed depends on the tension T and the linear mass density μ (mass per unit length): v = √(T/μ). Higher tension stiffens the string, pulling particles back faster and increasing speed. A heavier string increases inertia, slowing propagation down.
- In sound: The speed depends on the medium's density and bulk modulus (elasticity). In air, sound speed increases with temperature because faster-moving molecules transmit collisions quicker.
Boundary Reflection & Phase Changes
When a wave reaches the boundary of its medium, it undergoes reflection. The phase of the reflected wave depends on the boundary's rigidity:
- Fixed End Reflection: If the end of the medium is rigidly clamped, it cannot move. When a crest arrives, it exerts an upward force on the support. The support exerts an equal and opposite downward reaction force on the string (Newton's Third Law), causing the reflected wave to return **inverted** (a phase shift of 180° or π radians).
- Free End Reflection: If the end of the medium can move freely (such as a string attached to a frictionless ring on a post), the ring overshoots. The wave reflects **upright** without any inversion or phase change.
- Absorptive / Infinite End: If the medium extends indefinitely or terminates in a damping absorber, all energy is transmitted or absorbed, and no reflection occurs.
Solved Examples
A wave propagates along a stretched string. The distance between two consecutive wave crests is measured to be 0.40 meters. If the wave source vibrates at a frequency of 50 Hz, calculate (a) the wavelength of the wave, and (b) the wave speed along the string.
View Step-by-Step Solution
- Identify the given values: frequency f = 50 Hz, and distance between consecutive crests is the definition of wavelength.
- Part (a): By definition, the distance between two consecutive crests is one full wavelength (λ).
- Therefore, wavelength λ = 0.40 meters.
- Part (b): Use the wave speed equation: v = f · λ.
- Substitute values: v = 50 Hz · 0.40 m = 20.0 m/s.
- The wavelength is 0.40 m and the wave speed is 20.0 m/s.
**Final Answer:** λ = 0.40 m, v = 20.0 m/s
A wave pulse travels down a 6.0-meter-long string with a speed of v = 30 m/s. The pulse hits a fixed support at the end of the string. (a) How long does it take for the pulse to travel to the boundary and return to the starting point? (b) Describe the shape of the pulse upon reflection.
View Step-by-Step Solution
- Identify variables: string length L = 6.0 m, wave speed v = 30 m/s.
- Part (a): The total travel distance for a round trip (to the end and back) is d = 2 · L = 2 · 6.0 = 12.0 meters.
- Use the speed formula: t = d / v.
- Substitute values: t = 12.0 m / 30 m/s = 0.40 seconds.
- Part (b): Because the boundary is a fixed support, the boundary exerts an equal and opposite reaction force on the string as the pulse arrives (Newton's Third Law).
- This reaction force causes the reflected pulse to invert, resulting in a 180° (π radians) phase change.
- The round trip time is 0.40 seconds, and the pulse reflects inverted.
**Final Answer:** t = 0.40 s, inverted reflection (phase shift of π)
A sound wave traveling in air at a speed of v = 340 m/s has a wavelength of 0.85 meters. (a) Find the frequency of the sound wave. (b) If the sound wave enters a pool of water where its speed increases to v_water = 1480 m/s, what is the new wavelength of the sound wave? (Note: The frequency of a wave is determined by the source and does not change when entering a new medium).
View Step-by-Step Solution
- Identify variables: initial speed v = 340 m/s, wavelength λ = 0.85 m.
- Part (a): Solve for frequency using the wave equation: f = v / λ.
- Substitute values: f = 340 / 0.85 = 400 Hz.
- Part (b): In water, the frequency remains f = 400 Hz, but the speed increases to v_water = 1480 m/s.
- Use the wave equation solved for wavelength: λ_water = v_water / f.
- Substitute values: λ_water = 1480 / 400 = 3.70 meters.
- The frequency of the sound is 400 Hz, and its wavelength in water is 3.70 meters.
**Final Answer:** f = 400 Hz, λwater = 3.70 m
Common Misconceptions & Pitfalls
- Misconception: Particles in a wave travel from the source to the receiver.
**Reality:** No. Particles only oscillate back and forth about their own fixed equilibrium positions. It is the disturbance, representing energy and momentum, that travels continuously down the medium. - Misconception: Increasing wave frequency increases its speed.
**Reality:** No. Pushing a source faster (higher frequency) decreases the wavelength proportionally, according to λ = v/f, keeping the wave speed v constant. Speed is independent of frequency and is set by medium tension and density. - Misconception: A wave pulse reflects upright from a solid wall.
**Reality:** The wall forces the boundary displacement to remain exactly zero. This constraint requires the wall to exert a reaction force that flips the pulse upside-down, creating an inverted reflection.
Practice Questions
Question 1
What is the physical significance of the wave equation v = f λ, and why does wave speed only depend on the medium?
Show Explanation
The wave equation v = f λ is a fundamental relation stating that the speed of a wave is the rate at which energy is transported through space. The speed v is determined solely by the mechanical properties of the medium (such as elasticity, tension, and mass density) because these properties dictate how fast adjacent particles pull on each other to propagate the disturbance. The frequency f is dictated entirely by the external source. Therefore, when the frequency changes, the wavelength λ adjusts inversely (λ = v/f) to maintain the constant speed allowed by the medium.
Question 2
Explain the microscopic mechanism behind fixed-end and free-end reflections in a string.
Show Explanation
Fixed-end reflection: The string end is anchored to a rigid boundary. When an upward pulse reaches the end, the string pulls up on the wall. According to Newton's Third Law, the wall pulls down on the string with an equal, opposite force. This downward pull creates a downward-oriented pulse that propagates back in the opposite direction (inverted, 180° phase shift). Free-end reflection: The string is attached to a ring that slides frictionless on a rod. As the pulse arrives, the ring moves upward freely. Because of inertia and lack of downward resistance, the ring overshoots to twice the pulse amplitude. The tension in the string then pulls the ring back down, launching an upright pulse back along the string (no inversion, 0° phase shift).
Question 3
What is a wave packet or wave pulse, and how does it differ from a continuous harmonic wave?
Show Explanation
A wave pulse is a single, isolated disturbance that travels through a medium (e.g., a single snap of a string), having a definite beginning and end. A continuous harmonic wave is a repeating sequence of wave pulses generated by a continuously oscillating source (e.g., a motor driving a string up and down permanently), described by a sinusoidal function. While a continuous wave has a single defined frequency, a wave pulse contains a broad spectrum of frequencies superposed together.
Question 4
Explain why mechanical waves cannot travel through a vacuum, whereas electromagnetic waves can.
Show Explanation
Mechanical waves require a physical medium because they propagate through particle-to-particle interactions. The energy of the disturbance is passed along via mechanical elastic forces (like tension, pressure, or shear) acting between neighboring molecules. In a vacuum, there are no particles to exert forces or transfer momentum. In contrast, electromagnetic waves consist of self-propagating, oscillating electric and magnetic fields that do not rely on atomic interactions and can exist in empty space.
Frequently Asked Questions
- What is a wave?
- A wave is a disturbance that propagates through space and time, transferring energy and momentum from one location to another without transporting physical matter.
- What is a mechanical wave?
- A mechanical wave is a wave that requires a physical substance (solid, liquid, or gas) as a medium to travel through. Examples include sound waves, water waves, and seismic waves.
- What is the difference between transverse and longitudinal waves?
- In transverse waves, the medium particles vibrate perpendicular to the direction of wave travel (e.g., string waves). In longitudinal waves, particles vibrate parallel to the direction of wave travel (e.g., sound waves).
- Does the speed of a wave depend on its amplitude?
- No. For linear mechanical waves, the speed is determined solely by the mechanical properties of the medium (such as tension and mass density) and is independent of the wave's amplitude or frequency.
- What is a crest and a trough?
- A crest is the point of maximum positive displacement in a transverse wave. A trough is the point of maximum negative displacement.
- What is a compression and a rarefaction?
- In a longitudinal wave, a compression is a region where the medium particles are squeezed closest together (high pressure/density). A rarefaction is a region where the particles are spread furthest apart (low pressure/density).